The following problem looks to be classical, but I fail to find a reference for it. If you know it, please help me.

Given a bounded measurable function $h:{\mathbb R}_+\to\mathbb R$ and a number $a\ge0$, find a Lipschitz function $u$ satisfying $$u\ge0,\qquad u'\ge h,\qquad u(u'-h)=0.$$ That is, one of both constraints is saturated.

It seems to me that the solution is unique, and that $$u(T)=\min\{p(T)\,|\,p\ge0,\,p'\ge h\}.$$ Also, there seems to be a dynamic programming principle in the following sense: if $0<S<T$, then finding $u(T)$ amounts to finding $u(S)$ and then solving the same problem but replacing $t=0$ by $t=S$ and the data $a$ by $u(S)$.

The following problem looks to be classical, but I fail to find a reference for it. If you know it, please help me.

Given a bounded measurable function $h:{\mathbb R}_+\to\mathbb R$ and a number $a\ge0$, find a Lipschitz function $u$ satisfying $$u\ge0,\qquad u'\ge h,\qquad u(u'-h)=0,\qquad u(0)=a.$$ That is, one of both constraints is saturated.

It seems to me that the solution is unique, and that $$u(T)=\min\{p(T)\,|\,p\ge0,\,p'\ge h\}.$$ Also, there seems to be a dynamic programming principle in the following sense: if $0<S<T$, then finding $u(T)$ amounts to finding $u(S)$ and then solving the same problem but replacing $t=0$ by $t=S$ and the data $a$ by $u(S)$.